Optimal. Leaf size=94 \[ -\frac{2}{21} (2 x+3)^{7/2}+\frac{62}{45} (2 x+3)^{5/2}+\frac{526}{81} (2 x+3)^{3/2}+\frac{3278}{81} \sqrt{2 x+3}+12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{4250}{81} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
[Out]
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Rubi [A] time = 0.28198, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{2}{21} (2 x+3)^{7/2}+\frac{62}{45} (2 x+3)^{5/2}+\frac{526}{81} (2 x+3)^{3/2}+\frac{3278}{81} \sqrt{2 x+3}+12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{4250}{81} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
[In] Int[((5 - x)*(3 + 2*x)^(7/2))/(2 + 5*x + 3*x^2),x]
[Out]
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Rubi in Sympy [A] time = 48.8042, size = 83, normalized size = 0.88 \[ - \frac{2 \left (2 x + 3\right )^{\frac{7}{2}}}{21} + \frac{62 \left (2 x + 3\right )^{\frac{5}{2}}}{45} + \frac{526 \left (2 x + 3\right )^{\frac{3}{2}}}{81} + \frac{3278 \sqrt{2 x + 3}}{81} - \frac{4250 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{243} + 12 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5-x)*(3+2*x)**(7/2)/(3*x**2+5*x+2),x)
[Out]
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Mathematica [A] time = 0.0864213, size = 112, normalized size = 1.19 \[ -\frac{2}{21} (2 x+3)^{7/2}+\frac{62}{45} (2 x+3)^{5/2}+\frac{526}{81} (2 x+3)^{3/2}+\frac{3278}{81} \sqrt{2 x+3}-6 \log \left (1-\sqrt{2 x+3}\right )+6 \log \left (\sqrt{2 x+3}+1\right )-\frac{4250}{81} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((5 - x)*(3 + 2*x)^(7/2))/(2 + 5*x + 3*x^2),x]
[Out]
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Maple [A] time = 0.017, size = 80, normalized size = 0.9 \[ -{\frac{2}{21} \left ( 3+2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{62}{45} \left ( 3+2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{526}{81} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{3278}{81}\sqrt{3+2\,x}}-6\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{4250\,\sqrt{15}}{243}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }+6\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5-x)*(3+2*x)^(7/2)/(3*x^2+5*x+2),x)
[Out]
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Maxima [A] time = 0.798576, size = 131, normalized size = 1.39 \[ -\frac{2}{21} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{62}{45} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{526}{81} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{2125}{243} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{3278}{81} \, \sqrt{2 \, x + 3} + 6 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x + 3)^(7/2)*(x - 5)/(3*x^2 + 5*x + 2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.289831, size = 134, normalized size = 1.43 \[ -\frac{1}{8505} \, \sqrt{3}{\left (8 \, \sqrt{3}{\left (270 \, x^{3} - 738 \, x^{2} - 8639 \, x - 24728\right )} \sqrt{2 \, x + 3} - 17010 \, \sqrt{3} \log \left (\sqrt{2 \, x + 3} + 1\right ) + 17010 \, \sqrt{3} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 74375 \, \sqrt{5} \log \left (\frac{\sqrt{3}{\left (3 \, x + 7\right )} - 3 \, \sqrt{5} \sqrt{2 \, x + 3}}{3 \, x + 2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x + 3)^(7/2)*(x - 5)/(3*x^2 + 5*x + 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 138.764, size = 138, normalized size = 1.47 \[ - \frac{2 \left (2 x + 3\right )^{\frac{7}{2}}}{21} + \frac{62 \left (2 x + 3\right )^{\frac{5}{2}}}{45} + \frac{526 \left (2 x + 3\right )^{\frac{3}{2}}}{81} + \frac{3278 \sqrt{2 x + 3}}{81} + \frac{21250 \left (\begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 > \frac{5}{3} \\- \frac{\sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 < \frac{5}{3} \end{cases}\right )}{81} - 6 \log{\left (\sqrt{2 x + 3} - 1 \right )} + 6 \log{\left (\sqrt{2 x + 3} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5-x)*(3+2*x)**(7/2)/(3*x**2+5*x+2),x)
[Out]
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GIAC/XCAS [A] time = 0.275098, size = 136, normalized size = 1.45 \[ -\frac{2}{21} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + \frac{62}{45} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{526}{81} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{2125}{243} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) + \frac{3278}{81} \, \sqrt{2 \, x + 3} + 6 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 6 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x + 3)^(7/2)*(x - 5)/(3*x^2 + 5*x + 2),x, algorithm="giac")
[Out]